SMALL GROUP I - Part of the Jean Morlet Chair held by François Lalonde and Andrei Teleman
Recent Challenges in Contact Geometry
Dates: 8-12 June 2015 at CIRM (Marseille, France)
Contact geometry is now developing at a very high pace. Contact manifolds play fundamental roles in low dimensional topology (since all 3-manifolds admit a contact structure) and in all dimensions where they are the odd dimensional ana- logue of symplectic manifolds. Although the subject of contact geometry will be also addressed somewhere else in this proposal, especially in relation with Gauge
theory, it is certainly worth exploring issues that connect Contact geometry with
CR-geometry and with hard techniques in Symplectic topology.
Recent results of Margherita (Sheila) Sandon concern a bi-invariant metric on the
group of contactomorphisms of an arbitrary contact manifold. This subject is related to the theory of orderability launched by Eliashberg and Polterovich. It also relates to the contact version of the Non-squeezing theorem, a very subtle phenomenon. She has shown that the rigidity behaviours in contact geometry can be understood through the so-called translated points. She is now working on a programme that will involve a contact version of the Arnold conjecture and the right Floer contact homologies. If it works out, this would be a major progress in the field and could be one of the foci of the workshop.
On the other hand, recent results of Vincent Colin, Paolo Ghiggini and Ko Honda
give reasonable hope to understand the structure of some contact manifolds in terms of open book decompositions, relating the structure to the qualitative properties of the dynamics of the Reeb flow. A useful tool in this is the very recent proof (or ongoing proof) of the equivalence of the Heegard-Floer homology and the Embedded Contact Homology. Results have also been obtained on the fillable or non-fillable properties of contact manifolds - which is the bridge between the contact world and the symplectic one (a contact manifold is llable if it the boundary of a symplectic manifold whose symplectic structure is compatible with the contact structure on the boundary). Finally, Hutchings obtained sharp lower or upper bounds on the values of some basic capacities using ECH. In the light of the equivalence of HFH and ECH, this could pave the way to a better understanding of contact capacities.
Preliminary list of speakers